We discuss the important class of continuous functions of one variable. We teach you how to prove that a function is continuous. We also provide several examples and exercises with detailed answers to illustrate our course. To fully understand the content of this page, some tools on the limit of functions are necessary. Generalities on … Read more
Eigenvalues of a matrix are a fundamental concept in linear algebra that provide valuable information about the behavior of matrices. They play a crucial role in various fields, including physics, engineering, data analysis, and computer science. In this article, we will explore the concept of eigenvalues, understand their significance, and delve into methods for finding … Read more
In this article, we propose an exercise that describes the optimal method of the step gradient for coercive functions on spaces of finite dimension. These specific problems are used in optimization theory and some applications in finance. We mention also we can use the convex functions to study the optimality of some models. A problem … Read more
We will delve into the concept of a hyperplane linear subspace, explore its properties, and examine its applications in different domains. Hyperplanes are essential concepts in linear algebra that provide a framework for understanding linear subspaces in multidimensional spaces. They play a crucial role in various fields, including geometry, machine learning, and optimization. What is … Read more
We offer elementary probability exercises at the level of high school. We show the student how to use the formula of compound probabilities, how to use conditional probability, and the Bayes theorem. What is a random experiment? To study phenomena, it is important to carry out experiments. In fact, depending on the situation, certain phenomena … Read more
Riemann integral exercises with detailed answers are offered on this page. These kinds of integrals of bounded function in bounded intervals, almost everywhere continuous. Thus, all continuous or monotone functions are Riemann integrable. Lower integral, upper integral, and proper integral In this section, we give a concise definition of Riemann integrals. To this end, we … Read more
We discuss the properties of the translation operator defined in a Lebesgue space. The latter appears in the study of differential equations, mainly equations with negative memories called also delay equations. The semigroup property of the translation operator Let $p\ge 1$ be a real number and $E=L^p([0,+\infty)$ be the space of all measurable functions such … Read more
Real numbers are the backbone of mathematics, spanning a wide range of values along the number line. They’re like the superheroes of numbers, including familiar integers, fractions, and decimals, both rational and irrational. But did you know that the properties of real numbers are actually used in real life? That’s right, these numbers represent quantities, … Read more
Linear transformations and matrices form the backbone of numerous mathematical and scientific disciplines, ranging from computer graphics and machine learning to physics and engineering. The interplay between these concepts is not only intellectually stimulating but also crucial in solving real-world problems efficiently. In this article, we will explore the deep connection between linear transformations and … Read more
We discuss the Lyapunov stability for nonlinear systems. Indeed, having a reference solution, a stationary solution, one wonders if the ODE solution is closed to this reference when the time is very large. Throughout this post, we suppose that $F:\Omega\to \mathbb{R}^d$ is a continuously differentiable function, $C^1(\Omega)$, where $\Omega$ is an open set of $\mathbb{R}^d$. … Read more
Matrix operations are mathematical operations performed on matrices, which are rectangular arrays of numbers. Matrices are widely used in various fields, including mathematics, physics, computer science, and engineering. What is a matrix? Formally, a matrix is ββjust an array in which we put entries. This table can be a square or a rectangle. Mathematically, a … Read more
Primitives of continuous functions play a key role in the calculation of integrals. Therefore, in this article, we will use examples to introduce a simple technique for determining continuous function primitives. We assume that the reader knows how to calculate integrals. Generalities on primitives of continuous functions Let $f:D\subset \mathbb{R}$ be a continuous function. The … Read more